Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C83 · Schur multiplier (H2) = C25 |
| direct product | C83×#17423 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C83 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C83 |
| Frattini, cosocle | C22×C4 | C24×C83 |
| prime radical | C22×D4×C83 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C83 ◁ C2×C4×Q8×C83 ◁ G | |
922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 4 generators on 115 points:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(19,25)(22,28)(26,31)(29,32)4565 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 10411);